Thanks to the discreteness of the Dirichlet process, the latent parameters \(\theta _i\) 's exhibit ties with positive probability. Hence, the Dirichlet process mixture model is also routinely used to perform clustering since it partitions observations into groups based on whether their corresponding latent parameters \(\theta _i\)  coincide or not. The ubiquitous use of Dirichlet process mixtures for clustering motivates the interest in the asymptotic behaviour of the posterior distribution of the underlying partition, and in particular in
the inferred number of clusters (i.e. subpopulations), as the number of observations increases. [1]} showed posterior consistency of the mixing distribution \(\tilde{P}\)  under general conditions.
However, this does not imply consistency for the number of clusters, due to the use of the Wasserstein distance. Indeed,
[2]} proved that Dirichlet process mixtures are not consistent for the number of components when data are generated from a mixture with a single standard normal component.
See also [3]} for extensions.
These results, however, are derived under the assumption that the concentration parameter \(\alpha \)  is known and fixed.
This is crucial because the clustering behaviour of Dirichlet process mixtures is governed by the choice of \(\alpha \) .
Indeed, under the Dirichlet process mixture model, the prior probability of observing ties is a function solely of \(\alpha \) , since \(\text{pr}( \theta _i = \theta _j ) = 1/(\alpha +1)\) .
